**5. Results**

Numerical analysis of the problem in the process tomography takes place using, among others, finite element methods, finite difference methods, or boundary element methods. In the case of data shortages, we talk about fixed problems and in the case of excess with overdetermined problems. Automatic data analysis is an important part of the diagnosis of the process based on tomography. Knowledge of the process can make image reconstruction more resistant to incomplete or corrupted data. As a result of the calculations, we obtain a reconstructed image, for example, acoustic impedance, permittivity or conductivity, and dielectric loss to the parameters of a physical process (phase embolism, density, type of concentration). Advanced analysis leads to the extraction of features.

The example of image detection is shown in **Figure 9**. The images show the image reconstruction achieved by the Gauss-Newton method. A mesh of finite elements has been generated inside the area. The algorithm is solving for such a distribution of conductivity, so that the values of interelectrode voltages calculated on its basis are as close as possible to the measuring values of these voltages.

**Figure 10** presents the image reconstruction by electrical capacitance tomography using the Levenberg-Marquardt method and the modified Levenberg-Marquardt method. Grids used in calculations: rare 2218 nodes and 4146 finite elements and dense 7861 nodes and 15,146 A Nondestructive Distributed Sensor System for Imaging in Industrial Tomography http://dx.doi.org/10.5772/intechopen.79567 35

attenuation coefficient, and its derivative through frequency allow after appropriate reconstructive transformations, imaging of the internal structure of the tested medium, as well as flow parameters such as its speed, average speed, or profile speed. The basis for imaging is differences in local values of specific acoustic parameters. The image obtained by means of appropriate reconstruction methods presents the distribution (obtained from the measurement data using the scanning technique from as many directions as possible after the

The problem of image construction in the case of ultrasound very often leads to the overdeter-

where *W* is the matrix of dimensions m x n and m > n, *s* is the right-hand side vector (one

where the last minimum is taken for all vectors *f* which fulfill the previous relation. Equation (8)

Numerical analysis of the problem in the process tomography takes place using, among others, finite element methods, finite difference methods, or boundary element methods. In the case of data shortages, we talk about fixed problems and in the case of excess with overdetermined problems. Automatic data analysis is an important part of the diagnosis of the process based on tomography. Knowledge of the process can make image reconstruction more resistant to incomplete or corrupted data. As a result of the calculations, we obtain a reconstructed image, for example, acoustic impedance, permittivity or conductivity, and dielectric loss to the parameters of a physical process (phase embolism, density, type of concentration).

The example of image detection is shown in **Figure 9**. The images show the image reconstruction achieved by the Gauss-Newton method. A mesh of finite elements has been generated inside the area. The algorithm is solving for such a distribution of conductivity, so that the values of interelectrode voltages calculated on its basis are as close as possible to the measur-

**Figure 10** presents the image reconstruction by electrical capacitance tomography using the Levenberg-Marquardt method and the modified Levenberg-Marquardt method. Grids used in calculations: rare 2218 nodes and 4146 finite elements and dense 7861 nodes and 15,146

*W f = s,* (8)

, ‖*f* ∗‖<sup>2</sup> = min ‖*f*‖<sup>2</sup> (9)

, which minimizes Euclidean

\*

ultrasonic pulses have passed through the tested environment) [22].

One of the ways of the solution (Eq. (7)) is to find the vector *f*

is well known as a linear least squares problem (LSP).

Advanced analysis leads to the extraction of features.

column matrix), and *f* is the solution vector.

‖*r*‖

34 New Trends in Industrial Automation

ing values of these voltages.

**5. Results**

mined algebraic set of equations that can be expressed in the matrix form:

norm of residual vector *r* for the known matrix *W* and vector s, and it means:

<sup>2</sup> = min ‖*s* − *Wf*‖<sup>2</sup>

**Figure 9.** Image reconstruction (ERT): (a) Gauss-Newton with Laplace regularization and (b) Gauss-Newton with Tikhonov regularization (ERT).

**Figure 10.** Example of the image reconstruction (ECT). (a) Levenberg-Marquardt method and (b) modified Levenberg-Marquardt method.

finite elements. The numerical experiment was carried out on noisy data. **Figure 11** presents the image reconstruction by elastic net method in ECT.

**Figure 12** shows images of the experiments by ultrasound tomography. The algorithm was designed in such a way that an overdetermined system of equations could be generated, i.e., one for which the number of equations is greater than the number of unknowns. A feature of the tomography is, among other things, that the coefficient matrix is a rectangular deficiency of the pseudo-rank matrix. In such cases, you should consider trial solutions and choose only one of them. The obtained results are a raw tomographic image for synthetic data. In the numerical experiments presented, no additional adjustment method was used to obtain

inverse problem, which allows the imaging of processes. In the presented solution, we move from mathematical formalism to determining, analyzing, verifying, and checking systems that monitor and control physical processes. The system benefits various economic and industrial sectors. Advanced tools allow to capture both cybernetic abstractions and system dynamics. The system employs a communication interface, unique optimization algorithms, and data analysis algorithms for image reconstruction and process monitoring. The use of systems based on elec-

A Nondestructive Distributed Sensor System for Imaging in Industrial Tomography

http://dx.doi.org/10.5772/intechopen.79567

37

trical and ultrasonic tomography can significantly improve industrial processes.

University of Economics and Innovation/Research and Development Center Netrix S.A,

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[3] Ostrowski KL, Luke SP, Williams RA. Simulation of the performance of electrical capacitance tomography for measurement of de6nse phase pneumatic conveying. Chemical

[4] Rybak G, Chaniecki Z, Grudzień K, Romanowski A, Sankowski D. Non–invasive methods of industrial process control, Informatyka, Automatyka. Pomiary w Gospodarce i

[5] Beck MS, Byars M, Dyakowski T, Waterfall R, He R, Wang SJ, Yang WQ. Principles and industrial applications of electrical capacitance tomography. Measurement and Control.

[6] Garbaa H, Jackowska-Strumiłło L, Grudzień K, Romanowski A. Simulation of gravitational solids flow process and its parameters estimation by the use of electrical capacitance tomography and artificial neural networks. Informatyka, Automatyka, Pomiary w

[7] Filipowicz SF, Rymarczyk T. The shape reconstruction of unknown objects for inverse

Address all correspondence to: tomasz@rymarczyk.com

**Author details**

Tomasz Rymarczyk

Lublin, Poland

**References**

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**Figure 11.** Image reconstruction by elastic net method (ECT): (a) model and (b) image reconstruction.

**Figure 12.** Localized on the 60 × 60 plane, tomographic images (UST).

images without streaks. The results obtained using the proposed method are a faithful representation of the modeled objects and enable their precise location in the considered area.
